Bring a large pot of generously salted water to a boil. Cook the spaghetti according to the package directions until just shy of al dente. Before draining, reserve at least 1 cup of the starchy pasta water. Drain the pasta and set aside.
While the pasta cooks, season the chicken pieces with salt and black pepper. Melt the butter in a large skillet over medium-high heat. Add the chicken and cook until golden brown and cooked through, about 6 to 8 minutes. Transfer the chicken to a plate and set aside.
Reduce the heat to medium. Add the garlic to the same skillet and cook for about 30 seconds, stirring constantly, until fragrant.
Add the chopped fresh tomatoes and cook for 3 to 5 minutes, stirring occasionally, until they begin to soften and release their juices.
Pour in the Sauvignon Blanc and use a wooden spoon to scrape up any browned bits from the bottom of the skillet. Simmer for 2 to 3 minutes to allow the wine to reduce slightly.
Return the cooked chicken to the skillet, then add the drained spaghetti. Toss everything together over medium-low heat.
Add the freshly grated Parmesan and ½ cup of the reserved pasta water. Toss continuously until the cheese melts and the pasta water combines with the butter, wine, tomato juices, and Parmesan to create a light, silky sauce that coats the spaghetti. Add more pasta water, a little at a time, as needed.
Continue tossing and cooking the pasta in the sauce for 1 to 2 minutes, until the spaghetti is al dente and the sauce has thickened slightly and clings to the noodles.
Remove the skillet from the heat and gently toss in the fresh basil. Taste and adjust the salt and black pepper as needed.
Serve immediately with plenty of additional freshly grated Parmesan and, preferably, the rest of that Sauvignon Blanc.